The Enquirer understands that Carillion has been given the bullet from the race leaving Lend Lease, Skanska and Vinci still in the hunt for the £250m contract at Worthy Down in Winchester.
The scheme – known as Project Wellesley – will take around five years to complete.
The new facility will bring together onto one site the majority of logistics and personnel administration training for the Royal Navy, Army and Royal Air Force.
It will provide training facilities and living accommodation for up to 2,000 students and staff from the Armed Forces.
The plan is for the centre to be up and running by late 2018, whilst maintaining training at the site throughout the construction period.
The development of the Worthy Down site will enable the relocation of military units from Princess Royal Barracks Deepcut in Surrey, which is then intended to be closed and sold.
It will also see the Defence Maritime Logistics School, currently at HMS Raleigh, in Cornwall, and the RAF Supply and Movements Training Wing, based at RAF Halton, near Aylesbury, relocating to Worthy Down.